I do not think that there was a moment when they separated: that is a time T
such that they were one before T and separate after. Mathematics apparently started as an independent discipline from physics. Look at the work of Aristotle titled "Physics" and the work of Lucretius (exposition of Leucippus/Democritus system). They belong to physics, and they are certainly very different from mathematics of that time.
Legendary Pythagoras was credited with discoveries in both physics (relation between the length of a string and the sound it produces) and mathematics (theorems in geometry and number theory).
The close merge was achieved in the work of Archimedes; he
practiced both mathematics and physics, but still in his writings this
separation is evident. He never confuses physical reasoning with mathematical proof. In fact in a famous passage of his "Method" he emphasizes this difference.
At the later times, the two things sometimes came closer together, sometimes separated. And this should not be obscured by the fact that sometimes
these two lines of inquiry were practiced by the same people.
The closest merge probably happened in 18th and early 19th centuries,
but still at that time there was pure mathematics and mathematical physics and experimental physics. (See the famous exchange between
Fourier and Jacobi on the goals of science).
This situation persists even now, at the time of high specialization: there are still many people on the boundary practicing both physics and mathematics. And there are other people who are either physicists or mathematicians but not both.
So my conclusion is that there was no separation point: two sciences were separate from the beginning, but most of the time there was a close interaction between them.